On this week's recommended website MathOverFlow.com, Michael Hardy posed a question that elicited some helpful responses. Also, it included some interesting questions, which serve as this week's problems-to-solve.
When people graduate with honors from prestigious universities thinking everything in math is already known and the field consists of memorizing algorithms, then the educational system has failed in one of its major endeavors.
If members of the next freshman class will take just one one-semester math course before becoming the aforementioned graduates, here's what I think I might do.... I would not have a fixed syllabus of topics that the course must cover by the end of the semester. I would assign very simple but serious problems that I would not tell the students how to do. A few simple examples:
I said simple but serious, the latter meaning they will actually learn something worth learning about mathematics or about how to think about mathematics. Not all need be as elementary as these. With some of the less elementary problems I might sketch a solution or write out a solution in detail and then ask questions about the solution.
- 3x5 = 5+5+5 and 5x3 = 3+3+3+3+3. Why must this operation thus defined be commutative?
- A water lily has a single leaf floating on the surface of a pond. The leaf doubles in size every day. After 16 days it covers the whole pond. How long will it take two such leaves to cover the whole pond.... (See below for how one reader (B.A., OR) extended this problem)
- Visualize a square circumscribing a circle with radius 1. Here is how you use this to see that pi < 4 . [Explanation: Outside square has area of 4 and inside circle has area of pi.] Now figure out how to prove that pi > 3 by a similarly simple argument.
- Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, ...... Multiples of 18 are 18, 36, 54, 72, 90, ..... The smallest one that they have in common is 36. Multiples of 63 are 63, 126, 189, 252, 315, 378,..... Multiples of 77 are 77, 154, 231, 308, 385,.... Could this sequence go on forever without any number appearing in both lists? Is it the case that no matter which pair of numbers you start with, eventually some number will appear in both lists?
I would not fix in advance the date at which problems were to be turned in, but would set deadlines after discussion reveals that serious difficulties are overcome. I might also do some "teasing" concerning various math topics not covered.
B.A. (OR) offers these extensions of Water Lily Problem...
Sunday 2013-02-24 the MathNEXUS Problem of the Week included the following problem to ponder. : )
A water lily has a single leaf floating on the surface of a pond. The leaf doubles in size every day. After 16 days it covers the whole pond. How long will it take two such leaves to cover the whole pond?
We think that the intended answer is 15 days, but we have a few reservations.
How does the water lily double? Does its linear dimensions double? In that case, the area increases by a factor of 4. Or does its area double every day? We will assume the later.
We searched Bing for images of water lilies and found many. A water lily leaf is sort of oval with a chunk cut out of one end. The chunk is roughly triangular, but with curved sides.
Is the shape of the pond similar (ala geometry) to the shape of the water lily? Is the size of the pond 2^16 times the size of the water lily before it begins doubling? [Have we got that right?] Is the water lily positioned in exactly the right place in the pond before it begins doubling in size? If yes, then we can imagine the water lily doubling away until it covers the pond exactly.
What if the water lily is close to one side of the pond. Then after a few doublings, part of it would be in the pond and part of it would bump up on the land near that side. Then it would require more doubling to cover the pond.
But what if the shape of the pond is not similar to the shape of the water lily? Maybe it is square or rectangular or circular or perhaps irregular in shape as ponds are wont to be.
Now think about two water lilies. Are they side by side? End to end? At two random locations in the pond? And think again about the shape of the pond. Wherever the two water lilies are in the pond when they both begin doubling, might there be uncovered water between them? If their centers of mass remain in the original places, won't they overlap as they double?
Oops! What if the water lilies continue doubling after they cover the pond? How much territory will they cover in 17 days? 20 days? 100 days? In how many days will they cover the Earth?
Wow! This problem seems more like an investigation or a project. Thanks to MathNEXUS for posing such an interesting ponderable problem.
Hint: On Question #1, think about what commutivity means...and would it help to write 5+5+5 = 3+3+3+3+3?
On Question #2, Hardy writes: "Here lots of students say "8 days". I might warn them against that. This is the very hardest problem assigned in an algebra course that I taught, according to most of the students."
On question #3, assume that the circle has radius 2...then look at respective areas.
On Question #4, Hardy writes: "Usual answer: Yes. It will. Because 63 and 77 have nothing in common."
Solution Commentary: You are on your own....as described by Hardy, the poser of the questions.